Gradient index lens system

ABSTRACT

A radial gradient index lens system has a refractive index distribution. The refractive index decreases from an optical axis along a height. The lens system has two meniscus lens elements whose convex surfaces have a larger radius of curvature than concave surfaces. The meniscus lens elements are arranged symmetrically with respect to an aperture stop so that the convex surfaces thereof face an object side and an image side, respectively. The gradient index lens system is for use in a copying apparatus or as a reading optical system.

This is a continuation of application Ser. No. 08/073,753, filed Jun. 8, 1993, now abandoned.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a gradient index (GRIN) lens system in which two lenses are employed having a refractive index distribution from the optical axis along the height (i.e. radial refractive index distribution).

2. Description of the Prior Art

As a radial lens system having a refractive index distribution in a direction vertical to the optical axis and an axial lens system having a refractive index distribution along the optical axis, lens systems of comparatively small size such as objective lenses for optical discs have already been put into practical use. Lens systems of large size, however, have not yet been put into practical use because of the problem of the cost of manufacture.

Accordingly, radial and axial lens systems have hardly ever been used as optical systems for use in copying apparatuses which should be of large size and reading lens systems for use in image scanners. However, GRIN lens systems, which can be made compact by reducing the number of lens elements, are highly promising in the future. The following are some of the GRIN lens systems which have been proposed so far: (1) "Design of a Gradient-index Photographic Objective", Applied Optics Vol. 21, No. 6 shows an example of a design of a two-element photographic lens system; (2) U.S. Pat. No. 4,974,947 discloses two-element and three-element axial GRIN lens systems; (3) Japanese Laid-open Patent Application No. H2-285312 discloses a video and a photographic lens system in which at least one radial GRIN lens is used; and (4) "Design of an Optical System in Which a Radial GRIN Lens Is Used" (1991), Lectures at the 16th optics symposium, pages 39 to 40. All of them are photographic lens systems.

The lens system of (1) presents the problems of high cost and inferior processing capability since first and second lens elements thereof are very thick. Further, correction of chromatic aberration is not considered. The lens system of (2) is a wide angle lens system in which an axial GRIN lens is used having a refractive index distribution where the refractive index varies along the optical axis. Only data on a mono color are shown. In the lens system of (3), a radial GRIN lens is used mainly for correcting chromatic aberration. GRIN lens is applied to only one lens element of the photographic lens system including a plurality of lens elements. In the lecture of (4), achromatizing conditions required when the total power which is the sum of the powers of lens configuration and lens material is negative are obtained. As an application thereof, a method to obtain achromatizing conditions required when the total power is positive is shown. However, the example of the design is intended to reduce the total length of a zoom lens system.

On the other hand, for the reading lens systems for use in image scanners, the condition of chromatic aberration is strict compared to the photographic lens system. Accordingly, four to six lens elements are required when the lens system is the conventional spherical lens system of uniform configuration and material. Similarly, in the conventional lens systems for use in copying apparatuses having uniform refractive indices, at least three lens elements are required.

For the copying apparatuses and image scanners, reduction of size is strongly demanded, and accordingly, for the lens systems for use in those apparatuses, reduction of the number of lens elements and reduction of size are also strongly demanded.

SUMMARY OF THE INVENTION

An object of the present invention is to provide a GRIN lens system where the number of lens elements has largely been reduced, said GRIN lens being provided with performance required for lens systems for use in copying apparatuses and for image reading lens systems.

According to the first feature of the present invention, a radial GRIN lens system having a refractive index distribution where the refractive index decreases from the optical axis along the height is of a symmetrical type comprising two meniscus lens elements whose convex surfaces have a larger radius of curvature than concave surfaces, said meniscus lens elements being arranged symmetrically with respect to an aperture stop so that the convex surfaces thereof face an object side and an image side, respectively. The GRIN lens system fulfills the following conditions:

    ν0<ν1d

    0.20>ΣP>-0.05

where ν0 represents an axial Abbe number of the GRIN lens, ν1d is calculated by ##EQU1## where N1d represents a secondary gradient index coefficient of d-line, N1F represents a secondary gradient index coefficient of F-line and N1C represents a secondary gradient index coefficient of C-line, and ΣP represents a Petzval sum of the lens system.

According to the second feature of the present invention, a GRIN lens system includes two radial GRIN lenses each having a refractive index distribution from the optical axis along the height. In the two GRIN lenses, the power of lens configuration is negative and the power of lens material is positive, and the total power is positive.

According to the third feature of the present invention, in a GRIN lens system, with respect to at least one of two GRIN lenses, the power of lens configuration is negative and the power of lens material is positive, and the total power thereof is positive.

According to the fourth feature of the present invention, in a GRIN lens system, the power of lens configuration of a first lens element is negative and the power of lens material thereof is positive, and the power of lens configuration of a second lens element is positive and the power of lens material thereof is positive.

According to the fifth feature of the present invention, in a GRIN lens system, the power of lens configuration of a first lens element is positive and the power of lens material thereof is positive, and the power of lens configuration of a second lens element is negative and the power of lens material thereof is positive.

According to the sixth feature of the present invention, in a GRIN lens system, the power of lens configuration of a first lens element is negative and the power of lens material thereof is positive, and the power of lens configuration of a second lens is positive and the power of lens material thereof is negative.

BRIEF DESCRIPTION OF THE DRAWINGS

This and other objects and features of this invention will become clear from the following description, taken in conjunction with the preferred embodiments with reference to the accompanied drawings in which:

FIG. 1 is a cross-sectional view showing the lens arrangement of a first embodiment of the present invention;

FIGS. 2A to 2C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 1, respectively;

FIG. 3 is a cross-sectional view showing the lens arrangement of a second embodiment of the present invention;

FIGS. 4A to 4C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 3, respectively;

FIG. 5 is a cross-sectional view showing the lens arrangement of a third embodiment of the present invention;

FIGS. 6A to 6C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 5, respectively;

FIG. 7 is a cross-sectional view showing the lens arrangement of a fourth embodiment of the present invention;

FIGS. 8A to 8C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 7, respectively;

FIG. 9 is a cross-sectional view showing the lens arrangement of a fifth embodiment of the present invention;

FIGS. 10A to 10C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 9, respectively;

FIG. 11 is a cross-sectional view showing the lens arrangement of a sixth embodiment of the present invention;

FIGS. 12A to 12C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 11, respectively;

FIG. 13 is a cross-sectional view showing the lens arrangement of a seventh embodiment of the present invention;

FIGS. 14A to 14C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 13, respectively;

FIG. 15 is a cross-sectional view showing the lens arrangement of an eighth embodiment of the present invention;

FIGS. 16A to 16C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 15, respectively;

FIG. 17 is a cross-sectional view showing the lens arrangement of a ninth embodiment of the present invention;

FIGS. 18A to 18C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 17, respectively;

FIG. 19 is a cross-sectional view showing the lens arrangement of a tenth embodiment of the present invention;

FIGS. 20A to 20C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 19, respectively;

FIG. 21 is a cross-sectional view showing the lens arrangement of an eleventh embodiment of the present invention;

FIGS. 22A to 22C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 21, respectively;

FIG. 23 is a cross-sectional view showing the lens arrangement of a twelfth embodiment of the present invention;

FIGS. 24A to 24C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 23, respectively;

FIG. 25 is a cross-sectional view showing the lens arrangement of a thirteenth embodiment of the present invention;

FIGS. 26A to 26C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 25, respectively;

FIG. 27 is a cross-sectional view showing the lens arrangement of a fourteenth embodiment of the present invention;

FIGS. 28A to 28C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 27, respectively;

FIG. 29 is a cross-sectional view showing the lens arrangement of a fifteenth embodiment of the present invention; and

FIGS. 30A to 30C are graphic representations of spherical aberration, astigmatism and distortion of the lens system of FIG. 29, respectively.

BRIEF DESCRIPTION OF THE EMBODIMENTS

A radial GRIN lens system of the present invention is of a symmetrical type comprising two meniscus lens elements whose convex surfaces have a larger radius of curvature than concave surfaces, said meniscus lens elements being arranged symmetrically with respect to an aperture stop so that the convex surfaces thereof face an object side and an image side, respectively. The GRIN lens system fulfills the following conditions:

    ν0<ν1d                                               (1)

    0.20>ΣP>-0.05                                        (2)

where ν0 represents an axial Abbe number of the GRIN lens, ν1d is calculated by ##EQU2## where N1d represents a secondary gradient index coefficient of d-line, N1F represents a secondary gradient index coefficient of F-line and N1C represents a secondary gradient index coefficient of C-line, and ΣP represents a Petzval sum of the lens system.

For a copying apparatus where a lens system is used approximately at unity magnification, a symmetrical lens system has frequently been used. The advantage of the symmetrical lens system is that the designing is easy. This is because correction of only longitudinal aberrations has to be considered since lateral aberrations such as distortion, lateral chromatic aberration and coma are restrained. Moreover, it has been known that by using a radial GRIN lens having a gradient index distribution from the optical axis along the height, chromatic aberration can be corrected. In view of the above, in the present invention, with a completely symmetrical arrangement where two same radial GRIN lenses are used, axial chromatic aberration is also excellently corrected.

The GRIN lenses of the present invention are meniscus lenses whose convex surfaces have a larger radius of curvature than the concave surfaces thereof. When the surfaces are denoted by, from the convex surface on the left side, a first surface, a second surface, a third surface and a fourth surface, respectively, the first and fourth surfaces at the outside have positive power since the refractive power thereof gradually decreases from the optical axis along the height within the lens material, whereas the second and third surfaces at the inside have negative power. With an arrangement where an aperture stop S is arranged between the two lenses and the convex surfaces are arranged at the outside, the convex surfaces and the lens material produce positive power and the two concave surfaces produce negative power, which produces a power arrangement of positive, negative, positive as a whole. This arrangement is the same as those of what is called the triplet type lens system and the Gauss type lens system. Since the positive power is dispersed due to the convex surfaces and the lens material, aberration correction is excellently made.

Since the refractive indices vary along the height of the lens from the optical axis, the lens material will have a positive power when the indices of refraction increase from the optical axis to the outer edge of the lens and will have a negative power when the indices of refraction decrease from the optical axis to the outer edge.

It is well known that the correction expression of chromatic aberration is φS/ν0+φM/ν1d where φS represents the power of lens configuration, φM represents the power of lens material. In the lens systems of the present invention, since φS<0 and φM>0, the off-axial Abbe number is smaller than the axial Abbe number. Therefore, it is natural that ν1d>0. By fulfilling the condition (1), F-line and C-line are corrected with good balance. When the condition (1) is not fulfilled, the difference between the axial and off-axial Abbe numbers are so small that it is impossible to correct chromatic aberration.

In order to excellently correct field curvature, it is required that the Petzval sum ΣP fulfills the condition (2).

Subsequently, specific embodiments of the lens system for use in a copying apparatus using the GRIN lenses of the present invention will be described. FIGS. 1, 3, 5 and 7 are cross-sectional view showing the lens arrangements of first to fourth embodiments, respectively. Numerical data thereof are shown in Tables 1 to 4, respectively. In each table, radii of curvature ri of an ith surface from the object side and axial distances are shown. Since all of the lenses are GRIN lenses, axial refractive indices and Abbe numbers to d-line, and gradient index coefficients are also shown. Since the lens systems are completely symmetrical, data of only one lens element are shown.

The refractive index distribution is defined by

    N(r)=N0+N1r2+N2r4+N3r6+N4r8+ . . .                         (3)

where N(r) represents a refractive index at a height r from the optical axis, N0 represents an axial refractive index and Ni represents a gradient index coefficient.

FIGS. 2A to 2C, 4A to 4C, 6A to 6C and 8A to 8C are graphic representations of spherical aberration, astigmatism and distortion of the first to fourth embodiments. From the figures, it is understood that each aberration is extremely excellently corrected. The values of ν0, ν1d and ΣP of the conditions (1) and (2) for the first to fourth embodiments are shown in Table 16.

Subsequently, GRIN lens systems used as reading lens systems for use in image scanners will be described in accordance with five implementations thereof. In a first implementation, the power of lens configuration of each of the two radial GRIN lenses is negative and the power of lens material thereof is positive. The total power is positive. In a second implementation, with respect to at least one of two GRIN lenses, the power of lens configuration is negative and the power of lens material thereof is positive. The total power is positive. In a third implementation, the power of lens configuration of a first lens element is negative and the power of lens material thereof is positive, and the power of lens configuration of a second lens element is positive and the power of lens material thereof is positive. In a fourth implementation, the power of lens configuration of a first lens element is positive and the power of lens material thereof is positive, and the power of lens configuration of a second lens element is negative and the power of lens material thereof is positive. In a fifth implementation, the power of lens configuration of a first lens element is negative and the power of lens material thereof is positive, and the power of lens configuration of a second lens element is positive and the power of lens material is negative.

Hereinafter, specific embodiments of the above-mentioned first to fifth implementations will be described.

As the first implementation, fifth and sixth embodiments will be described. The lens arrangements thereof are shown in FIGS. 9 and 11, respectively. These embodiments comprise a first meniscus lens element convex to the object side and a second meniscus lens element convex to the image side. In both the first and second meniscus lens elements, the power of lens configuration is negative and the power of lens material is positive, and the total power of each lens element is positive. Regarding the power of lens material, the refractive index decreases from the optical axis along the height. Specifically, the first convex surface r1 from the object side and the lens material produce positive power, the two concave surfaces r2 and r3 at the inside produce negative power, and the most image side surface r4 produces positive power, which produces a power arrangement of positive, negative, positive as a whole. This arrangement is the same as those of what is called the triplet type lens system and the Gauss type lens system. Since the positive power is dispersed due to the convex surface and the lens material, aberration correction is excellently made.

Numerical data of the fifth and sixth embodiments are shown in Tables 5 and 6. Spherical aberration, astigmatism and distortion thereof are shown in FIGS. 10A to 10C and 12A to 12C. The refractive power distribution is generally defined by the previously-described equation (3).

In each table, N10 represents an axial refractive index of the first lens element to d-line, N20 represents an axial refractive index of the second lens element to d-line, ν10 represents an axial Abbe number of the first lens element, ν20 represents an axial Abbe number of the second lens element, N2d represents a secondary gradient index coefficient to d-line, N4d represents a quartic gradient index coefficient to d-line, N2F represents a secondary gradient index coefficient to F-line, and N2C represents a secondary gradient index coefficient to C-line.

Regarding axial chromatic aberration, it is required that both the first and second lens elements fulfill the following condition:

    νi0<νi'                                              (4)

where νi0 represents an axial Abbe number of an ith lens element and νi' is calculated by ##EQU3## where N_(2d) ^(i) represents a secondary gradient index coefficient of the ith lens element to d-line, N_(2F) ^(i) represents a secondary gradient index coefficient of the ith lens element to F-line and N_(2C) ^(i) is a secondary gradient index coefficient of the ith lens element to C-line.

The condition (4) is a condition required for excellently correcting mainly axial chromatic aberration. When this condition is not fulfilled, it is difficult to correct chromatic aberration.

The fifth and sixth embodiments also fulfill the following conditions: ##EQU4## where f1 represents a focal length of the ith lens element (total of the lens configuration and lens material), ψiS represents the power of lens configuration of the ith lens element, ψiM represents the power of lens material of the ith lens element, and ΣP represents a Petzval sum.

The condition (5) is a condition for appropriately determining the powers of the first and second lens elements. When the upper limit or the lower limit thereof is exceeded, the aberration correcting load on one of the lens elements increases, so that it is impossible to sufficiently correct aberrations. The condition (6) is a condition for excellently correcting field curvature. When the Petzval sum exceeds the upper limit of the condition (6), field curvature is undercorrected. When the Petzval sum exceeds the lower limit thereof, field curvature is overcorrected. The condition (7) is a condition required for appropriately determining the ratio of the power of lens configuration to the power of lens material of each of the first and second lens elements. When the upper limit of the condition (7) is exceeded, chromatic aberration is overcorrected. When the lower limit thereof is exceeded, chromatic aberration is undercorrected, so that it is impossible to sufficiently correct coma.

Values in connection with the conditions (4) to (7) for the fifth and sixth embodiments are shown in Table 17.

As the second implementation, seventh and eighth embodiments will be described. The lens arrangements thereof are shown in FIGS. 13 and 15, respectively. These embodiments comprise a first meniscus lens element convex to the object side and a second meniscus lens element convex to the image side. In the first lens element, the power of lens configuration is positive and the power of lens material is negative, and the total power of this lens element is positive. In the second lens element, the power of lens configuration is negative and the power of lens material is positive, and the total power of this lens element is positive. Since the power of lens configuration and the power of lens material between the first and second lens elements are opposite, aberration correction is excellently made.

Numerical data of the seventh and eighth embodiments are shown in Tables 7 and 8. Spherical aberration, astigmatism and distortion thereof are shown in FIGS. 14A to 14C and 16A to 16C. Similarly to the first implementation, the refractive power distribution is generally defined by the equation (3).

Regarding axial chromatic aberration, it is required that both the first and second lens elements fulfill the following condition:

    0<νi'<ν10                                            (8)

where:

νi' is calculated by ##EQU5## where N_(2d) ^(i) represents a secondary gradient index coefficient of the ith lens element to d-line, N_(2F) ^(i) represents a secondary gradient index coefficient of the ith lens element to F-line and N_(2C) ^(i) is a secondary gradient index coefficient of the ith lens element to C-line; and

ν10 represents an axial Abbe number of the first lens element.

The condition (8) is required to restrain, particularly, axial chromatic aberration by setting the Abbe number within the range defined by the condition (8). When the condition (8) is not fulfilled, it is impossible to correct chromatic aberration.

The seventh and eighth embodiments also fulfill the following conditions: ##EQU6## where f represents a focal length of the entire lens system (total of the lens configuration and lens material), f1 represents a focal length of the first lens element (total of the lens configuration and lens material), f2 represents a focal length of the second lens element (total of the lens configuration and lens material), ψ2S represents the power of lens configuration of the second lens element, ψ2M represents the power of lens material of the second lens element, and ΣP represents a Petzval sum.

The condition (9) is a condition for determining the power of the second lens element. When the upper limit thereof is exceeded, the power of lens configuration is so large that aberration generation amount increases. As a result, spherical aberration and astigmatism are overcorrected. In order to solve this problem, it is necessary to further increase the refractive power difference, which increases the manufacturing load. Such lens systems are of no practical use. When the lower limit of the condition (9) is exceeded, the power of lens material is so large that chromatic aberration is overcorrected. The condition (10) is a condition for appropriately determining the powers of lens configuration and lens material of the second lens element. When the upper limit thereof is exceeded, the power of lens material is large compared to the power of lens configuration, so that chromatic aberration is overcorrected. When the lower limit thereof is exceeded, the power of lens configuration is large, so that it is impossible to sufficiently correct coma. The condition (11) is a condition required for correcting field curvature. When the upper limit thereof is exceeded, field curvature is undercorrected. When the lower limit thereof is exceeded, field curvature is overcorrected. The condition (12) is a condition for determining the powers of the first and second lens elements. When the upper limit thereof is exceeded, the aberration correcting load on the second lens element increases, so that it is impossible to sufficiently correct aberrations. When the lower limit thereof is exceeded, the positive power of the first lens element increases, so that both spherical aberration and astigmatism are undercorrected.

Values in connection with the conditions (8) to (12) for the seventh and eighth embodiments are shown in Table 18.

As the third implementation, ninth and tenth embodiments will be described. The lens arrangements thereof are shown in FIGS. 17 and 19, respectively. These embodiments comprise a first meniscus lens element convex to the object side and a second meniscus lens element convex to the image side. In the first lens element, the power of lens configuration is negative and the power of lens material is positive, and the total power of this lens element is positive. In the second lens element, the power of lens configuration is positive and the power of lens material is positive, and the total power of this lens element is positive.

Numerical data of the ninth and tenth embodiments are shown in Tables 9 and 10. Spherical aberration, astigmatism and distortion thereof are shown in FIGS. 18A to 18C and 20A to 20C. Similarly to the previously-described implementations, the refractive power distribution is generally defined by the equation (3).

In the ninth and tenth embodiments, to correct axial chromatic aberration, it is required that the following conditions are fulfilled:

    νi0<νi'                                              (4)

    ν2'<0                                                   (13) ##EQU7## where νi0 represents an axial Abbe number of an ith lens element; νi' represents an Abbe number of the lens material calculated by ##EQU8## where N.sub.2d.sup.i represents a secondary gradient index coefficient of the ith lens element to d-line, N.sub.2F.sup.i represents a secondary gradient index coefficient of the ith lens element to F-line and N.sub.2C.sup.i is a secondary gradient index coefficient of the ith lens element to C-line.

The conditions (4) and (13) are conditions which the first and second lens elements are required to fulfill to excellently correct axial chromatic aberration. The condition (14) is a condition required for correcting axial chromatic aberration and lateral chromatic aberration with good balance as a whole. When the conditions (4) and (13) are fulfilled but the condition (14) is not fulfilled, both correction of axial chromatic aberration and correction of lateral chromatic aberration are difficult.

The ninth and tenth embodiments fulfill the following conditions: ##EQU9## where f represents a focal length of the entire lens system (total of the lens configuration and lens material), f1 represents a focal length of the first lens element (total of the lens configuration and lens material), f2 represents a focal length of the second lens element (total of the lens configuration and lens material), ψ2S represents the power of lens configuration of the second lens element, ψ2M represents the power of lens material of the second lens element, and ΣP represents a Petzval sum.

The condition (15) is a condition for determining the powers of the first and second lens elements. When the lower limit or the upper limit thereof is exceeded, the aberration correcting load on one of the lens elements increases, so that it is impossible to sufficiently correct aberrations. The condition (16) is a condition required for correcting field curvature. When the upper limit thereof is exceeded, field curvature is undercorrected. When the lower limit thereof is exceeded, field curvature is overcorrected. The condition (17) is a condition for appropriately determining the powers of lens configuration and lens material of the second lens element. When the upper limit thereof is exceeded, the power of lens material is small compared to that of the configuration, so that it is difficult to correct lateral chromatic aberration.

Values in connection with the conditions (4) and (13) to (17) for the ninth and tenth embodiments are shown in Table 19.

As the fourth implementation, eleventh to thirteenth embodiments will be described. The lens arrangements thereof are shown in FIGS. 21, 23 and 25, respectively. These embodiments comprise a first meniscus lens element convex to the object side and a second meniscus lens element convex to the image side. In the first lens element, both the power of lens configuration and the power of lens material are positive, and the total power of this lens element is positive. In the second lens element, the power of lens configuration is negative and the power of lens material is positive, and the total power of this lens element is positive.

Numerical data of the eleventh to thirteenth embodiments are shown in Tables 11, 12 and 13. Spherical aberration, astigmatism and distortion thereof are shown in FIGS. 22A to 22C, 24A to 24C and 26A to 26C. Similarly to the previously-described implementations, the refractive power distribution is generally defined by the equation (3).

It is required that these embodiments fulfill the following conditions: ##EQU10## where: ν20 represents an axial Abbe number of the second lens element;

ν2' is calculated by ##EQU11## where N2d represents a secondary gradient index coefficient of the second lens element to d-line, N2F represents a secondary gradient index coefficient of the second lens element to F-line and N2C represents a secondary gradient index coefficient of the second lens element to C-line;

f represents a focal length of the entire lens system (total of the lens configuration and lens material);

f1 represents a focal length of the first lens element (total of the lens configuration and lens material);

f2 represents a focal length of the second lens element (total of the lens configuration and lens material);

ψ2S represents the power of lens configuration of the second lens element;

ψ2M represents the power of lens material of the second lens element;

ΣP represents a Petzval sum.

ψ1S represents the power of lens configuration of the first lens element; and

ψ1M represents the power of lens material of the first lens element.

The condition (18) is a condition for correcting chromatic aberration. When the condition (18) is not fulfilled, it is impossible to correct chromatic aberration. The condition (19) is a condition for determining the power of the second lens element. When the upper limit thereof is exceeded, the power of lens configuration is so large that aberration generation amount increases. As a result, spherical aberration and astigmatism are overcorrected. In order to solve this problem, it is necessary to further increase the refractive power, which increases the manufacturing load. Such lens systems are of no practical use. When the lower limit of the condition (19) is exceeded, the power of lens material is so large that chromatic aberration is overcorrected. The condition (20) is a condition for determining the powers of lens configuration and lens material of the first lens element. When the upper limit thereof is exceeded, chromatic aberration is undercorrected. When the lower limit thereof is exceeded, the power of lens material is large, so that spherical aberration is overcorrected.

The condition (21) is a condition for appropriately determining the powers of lens configuration and lens material of the second lens element. When the upper limit thereof is exceeded, the power of lens material is large compared to the power of lens configuration, so that chromatic aberration is overcorrected. When the lower limit thereof is exceeded, the power of lens configuration is large, so that it is impossible to sufficiently correct coma. The condition (22) is a condition required for correcting field curvature. When the upper limit thereof is exceeded, field curvature is undercorrected. When the lower limit thereof is exceeded, field curvature is overcorrected. The condition (23) is a condition for determining the powers of the first and second lens elements. When the upper limit thereof is exceeded, the aberration correcting load on the second lens element increases, so that it is impossible to sufficiently correct aberrations. When the lower limit thereof is exceeded, the positive power of the first lens element increases, so that both spherical aberration and astigmatism are undercorrected.

Values in connection with the conditions (18) to (23) for the eleventh to thirteenth embodiments are shown in Table 20.

As the fifth implementations, fourteenth and fifteenth embodiments will be described. The lens arrangements thereof are shown in FIGS. 27 and 29, respectively. These embodiments comprise a first meniscus lens element convex to the object side and a second meniscus lens element convex to the image side. In the first lens element, the power of lens configuration is negative and the power of lens material is positive, and the total power of this lens element is positive. In the second lens element, the power of lens configuration is positive and the power of lens material is negative, and the total power of this lens element is positive. Since the powers of lens configuration and lens material of the first lens element and those of the second lens element are opposite, aberration correction is excellently made.

Numerical data of the fourteenth and fifteenth embodiments are shown in Tables 14 and 15. Spherical aberration, astigmatism and distortion thereof are shown in FIGS. 28A to 28C and 30A to 30C. Similarly to the previously-described implementations, the refractive power distribution is generally defined by the equation (3).

It is required that these embodiments fulfill the following conditions: ##EQU12##

The conditions (24) and (25) are conditions for correcting chromatic aberration. When these conditions are not fulfilled, it is impossible to correct chromatic aberration. The condition (26) is a condition for determining the powers of lens configuration and lens material of the second lens element. When the lower limit thereof is exceeded, spherical aberration is undercorrected, and it is impossible to sufficiently correct coma and chromatic aberration. When the upper limit thereof is exceeded, spherical aberration is overcorrected, and field curvature increases. The condition (27) is a condition for determining the powers of lens configuration and lens material of the first lens element. When the upper limit thereof is exceeded, spherical aberration is undercorrected. When the lower limit thereof is exceeded, spherical aberration is overcorrected, and it is impossible to sufficiently correct chromatic aberration.

The condition (28) is a condition for correcting field curvature. When the upper limit thereof is exceeded, field curvature is undercorrected. When the lower limit thereof is exceeded, field curvature is overcorrected. The condition (29) is a condition for determining the powers of the first and second lens elements. When the upper limit thereof is exceeded, the aberration correcting load on the second lens element increases, so that spherical aberration is undercorrected, and astigmatism is undercorrected. When the lower limit thereof is exceeded, the positive power of the first lens element increases, so that correction of chromatic aberration and correction of astigmatism are not well-balanced.

Values in connection with the conditions (24) to (29) for the fourteenth and fifteenth embodiments are shown in Table 21.

Obviously, many modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced other than as specifically described.

                  TABLE 1                                                          ______________________________________                                         (1st Embodiment)                                                               ______________________________________                                         f = 220.12   FNo. = 10.0                                                       Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    54.027                                                                                  d1     6.0                                                      r2    44.356                GRIN lens                                                         d2     9.0                                                      S    Aperture stop                                                                            d2'    9.0                                                      r3   -44.356                                                                                  d3     6.0   GRIN lens                                          r4   -54.027                                                                   ______________________________________                                         GRIN lens                                                                      N0 = 1.6910   ν0 = 54.8                                                     (Gradient index coefficients)                                                  d-line N1    -0.398640 × 10.sup.-3                                                                   N2  0.458588 × 10.sup.-7                            N3    -0.1142775 × 10.sup.-10                                     F-line N1    -0.400500 × 10.sup.-3                                                                   N2  0.460728 × 10.sup.-7                            N3    -0.114811 × 10.sup.-10                                      C-line N1    -0.397843 × 10.sup.-3                                                                   N2  0.457670 × 10.sup.-7                            N3    -0.114049 × 10.sup.-10                                      Condition ν.sub.1d = 150                                                    ______________________________________                                    

                  TABLE 2                                                          ______________________________________                                         (2nd Embodiment)                                                               ______________________________________                                         f = 200.1   FNo. = 10.0                                                        Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    42.247                                                                                  d1     4.0                                                      r2    35.027                GRIN lens                                                         d2     7.0                                                      S    Aperture stop                                                                            d2'    7.0                                                      r3   -35.027                                                                                  d3     4.0   GRIN lens                                          r4   -42.247                                                                   ______________________________________                                         GRIN lens                                                                      N0 = 1.5891   ν0 = 61.1                                                     (Gradient index coefficients)                                                  d-line N1    -0.616896 × 10.sup.-3                                                                   N2  0.172431 × 10.sup.-6                            N3    -0.425501 × 10.sup.-9                                                                   N4  0.241363 × 10.sup.-11                    F-line N1    -0.619774 × 10.sup.-3                                                                   N2  0.173236 × 10.sup.-6                            N3    -0.427486 × 10.sup.-9                                                                   N4  0.242490 × 10.sup.-11                    C-line N1    -0.615661 × 10.sup.-3                                                                   N2  0.172087 × 10.sup.-6                            N3    -0.424650 × 10.sup.-9                                                                   N4  0.240881 × 10.sup.-11                    Condition ν.sub.1d = 150                                                    ______________________________________                                    

                  TABLE 3                                                          ______________________________________                                         (3rd Embodiment)                                                               ______________________________________                                         f = 200.1   FNo. = 10.0                                                        Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    43.806                                                                                  d1     5.00                                                     r2    39.494                GRIN lens                                                         d2     6.50                                                     S    Aperture stop                                                                            d2'    6.50                                                     r3   -39.494                                                                                  d3     5.00  GRIN lens                                          r4   -43.806                                                                   ______________________________________                                         GRIN lens                                                                      N0 = 1.7440   ν0 = 44.9                                                     (Gradient index coefficients)                                                  d-line N2    -0.362601 × 10.sup.-3                                                                   N4  -0.709050 × 10.sup.-8                           N6     0.600650 × 10.sup.-10                                      F-line N2    -0.363507 × 10.sup.-3                                                                   N4  -0.710823 × 10.sup.-8                           N6     0.602152 × 10.sup.-10                                      C-line N2    -0.362212 × 10.sup.-3                                                                   N4  -0.708290 × 10.sup.-8                           N6     0.600007 × 10.sup.-10                                      Condition ν.sub.1d = 280                                                    ______________________________________                                    

                  TABLE 4                                                          ______________________________________                                         (4th Embodiment)                                                               ______________________________________                                         f = 200.0   FNo. = 8.0                                                         Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    60.645                                                                                  d1     9.0                                                      r2    53.608                GRIN lens                                                         d2     8.0                                                      S    Aperture stop                                                                            d2'    8.0                                                      r3   -53.608                                                                                  d3     9.0   GRIN lens                                          r4   -60.645                                                                   ______________________________________                                         GRIN lens                                                                      N0 = 1.8042   ν0 = 46.5                                                     (Gradient index coefficients)                                                  d-line N2    -0.195219 × 10.sup.-3                                                                   N4  -0.434268 × 10.sup.-8                           N6     0.229799 × 10.sup.-10                                      F-line N2    -0.195561 × 10.sup.-3                                                                   N4  -0.435028 × 10.sup.-8                           N6     0.230201 × 10.sup.-10                                      C-line N2    -0.195073 × 10.sup.-3                                                                   N4  -0.433943 × 10.sup.-8                           N6     0.229627 × 10.sup.-10                                      Condition ν.sub.1d = 400                                                    ______________________________________                                    

                  TABLE 5                                                          ______________________________________                                         (5th Embodiment)                                                               ______________________________________                                         f = 80.2   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    19.293                                                                                  d1     2.45  GRIN lens 1                                        r2    16.472                                                                                  d2     9.25                                                     r3   -23.466                                                                                  d3     2.70  GRIN lens 2                                        r4   -32.932                                                                            Σd = 14.40                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.7545   ν10 = 51.6                                                   (Gradient index coefficients)                                                  d-line N2d    -0.21795 × 10.sup.-2                                                                   N4d  -0.90912 × 10.sup.-7                           N6d     0.38546 × 10.sup.-8                                       F-line N2F    -0.21875 × 10.sup.-2                                                                   N4F  -0.91247 × 10.sup.-7                           N6F     0.38688 × 10.sup.-8                                       C-line N2C    -0.21760 × 10.sup.-2                                                                   N4C  -0.90768 × 10.sup.-7                           N6C     0.38485 × 10.sup.-8                                       GRIN lens 2                                                                    N20 = 1.6172   ν20 = 54.0                                                   (Gradient index coefficients)                                                  d-line N2d    -0.25582 × 10.sup.-2                                                                   N4d   0.43830 × 10.sup.-5                           N6d    -0.26498 × 10.sup.-8                                       F-line N2F    -0.25737 × 10.sup.-2                                                                   N4F   0.44096 × 10.sup.-5                           N6F    -0.26659 × 10.sup.-8                                       C-line N2C    -0.25515 × 10.sup.-2                                                                   N4C   0.43715 × 10.sup.-5                           N6C    -0.26429 × 10.sup.-8                                       ______________________________________                                    

                  TABLE 6                                                          ______________________________________                                         (6th Embodiment)                                                               ______________________________________                                         f = 80.5   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    19.181                                                                                  d1     2.45  GRIN lens 1                                        r2    16.217                                                                                  d2     9.91                                                     r3   -24.037                                                                                  d3     2.49  GRIN lens 2                                        r4   -36.661                                                                            Σd = 14.85                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.7130   ν10 = 53.9                                                   (Gradient index coefficients)                                                  d-line N2d    -0.21955 × 10.sup.-2                                                                   N4d  0.22363 × 10.sup.-6                            N6d     0.39555 × 10.sup.-8                                       F-line N2F    -0.22032 × 10.sup.-2                                                                   N4F  0.22441 × 10.sup.-6                            N6F     0.39693 × 10.sup.-8                                       C-line N2C    -0.21922 × 10.sup.-2                                                                   N4C  0.22329 × 10.sup.-6                            N6C     0.39495 × 10.sup.-8                                       GRIN lens 2                                                                    N20 = 1.5714   ν20 = 52.6                                                   (Gradient index coefficients)                                                  d-line N2d    -0.29765 × 10.sup.-2                                                                   N4d  0.68516 × 10.sup.-5                            N6d    -0.83169 × 10.sup.-8                                       F-line N2F    -0.29973 × 10.sup.-2                                                                   N4F  0.68995 × 10.sup.-5                            N6F    -0.83751 × 10.sup.-8                                       C-line N2C    -0.29675 × 10.sup.-2                                                                   N4C  0.68310 × 10.sup.-5                            N6C    -0.82919 × 10.sup.-8                                       ______________________________________                                    

                  TABLE 7                                                          ______________________________________                                         (7th Embodiment)                                                               ______________________________________                                         f = 80.0   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    25.000                                                                                  d1     3.58  GRIN lens 1                                        r2    49.999                                                                                  d2     7.57                                                     r3   -12.556                                                                                  d3     5.27  GRIN lens 2                                        r4   -25.844                                                                            Σd = 16.42                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.7545   ν10 = 51.6                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d     0.94985 × 10.sup.-3                                                                   N4d   0.17564 × 10.sup.-5                           N6d    -0.61095 × 10.sup.-8                                                                   N8d  -0.28143 × 10.sup.-10                    F-line                                                                               N2F     0.97644 × 10.sup.-3                                                                   N4F   0.18056 × 10.sup.-5                           N6F    -0.62806 × 10.sup.-8                                                                   N8F  -0.28931 × 10.sup.-10                    C-line                                                                               N2C     0.93845 × 10.sup.-3                                                                   N4C   0.17354 × 10.sup.-5                           N6C    -0.60362 × 10.sup.-8                                                                   N8C  -0.27805 × 10.sup.-10                    GRIN lens 2                                                                    N20 = 1.5891   ν20 = 61.1                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.23564 × 10.sup.-2                                                                   N4d   0.47788 × 10.sup.-5                           N6d    -0.89227 × 10.sup.-8                                                                   N8d   0.97269 × 10.sup.-10                    F-line                                                                               N2F    -0.23729 × 10.sup.-2                                                                   N4F   0.48122 × 10.sup.-5                           N6F    -0.89852 × 10.sup.-8                                                                   N8F   0.97950 × 10.sup.-10                    C-line                                                                               N2C    -0.23493 × 10.sup.-2                                                                   N4C   0.47644 × 10.sup.-5                           N6C    -0.88960 × 10.sup.-8                                                                   N8C   0.96977 × 10.sup.-10                    ______________________________________                                    

                  TABLE 8                                                          ______________________________________                                         (8th Embodiment)                                                               ______________________________________                                         f = 80.0   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    28.000                                                                                  d1     3.49  GRIN lens 1                                        r2    49.999                                                                                  d2     8.07                                                     r3   -12.786                                                                                  d3     4.90  GRIN lens 2                                        r4   -23.526                                                                            Σd = 16.45                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.7545   ν10 = 51.6                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d     0.58480 × 10.sup.-3                                                                   N4d   0.44491 × 10.sup.-6                           N6d    -0.51571 × 10.sup.-8                                                                   N8d  -0.28058 × 10.sup.-10                    F-line                                                                               N2F     0.60634 × 10.sup.-3                                                                   N4F   0.46130 × 10.sup.-6                           N6F    -0.53470 × 10.sup.-8                                                                   N8F  -0.29092 × 10.sup.-10                    C-line                                                                               N2C     0.57557 × 10.sup.-3                                                                   N4C   0.43788 × 10.sup.-6                           N6C    -0.50756 × 10.sup.-8                                                                   N8C  -0.27615 × 10.sup.-10                    GRIN lens 2                                                                    N20 = 1.5891   ν20 = 61.1                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.22997 × 10.sup.-2                                                                   N4d   0.38522 × 10.sup.-5                           N6d    -0.81238 × 10.sup.-8                                                                   N8d   0.97324 × 10.sup.-10                    F-line                                                                               N2F    -0.23150 × 10.sup.-2                                                                   N4F   0.38778 × 10.sup.-5                           N6F    -0.81780 × 10.sup.-8                                                                   N8F   0.97972 × 10.sup.-10                    C-line                                                                               N2C    -0.22931 × 10.sup.-2                                                                   N4C   0.38411 × 10.sup.-5                           N6C    -0.81006 × 10.sup.-8                                                                   N8C   0.97045 × 10.sup.-10                    ______________________________________                                    

                  TABLE 9                                                          ______________________________________                                         (9th Embodiment)                                                               ______________________________________                                         f = 80.9   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    40.160                                                                                  d1      3.34 GRIN lens 1                                        r2    22.988                                                                                  d2     14.09                                                    r3   -28.197                                                                                  d3      2.45 GRIN lens 2                                        r4   -26.458                                                                            Σd = 19.88                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.7725   ν10 = 49.8                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.28449 × 10.sup.-2                                                                   N4d   0.59569 × 10.sup.-5                           N6d    -0.70296 × 10.sup.-8                                        F-line                                                                               N2F    -0.28698 × 10.sup.-2                                                                   N4F   0.60090 × 10.sup.-5                           N6F    -0.70911 × 10.sup.-8                                        C-line                                                                               N2C    -0.28343 × 10.sup.-2                                                                   N4C   0.59346 × 10.sup.-5                           N6C    -0.70033 × 10.sup.-8                                        GRIN lens 2                                                                    N20 = 1.7545   ν20 = 51.6                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.99339 × 10.sup.-3                                                                   N4d  -0.79429 × 10.sup.-6                           N6d    -0.21551 × 10.sup.-8                                        F-line                                                                               N2F    -0.98607 × 10.sup.-3                                                                   N4F  -0.78844 × 10.sup.-6                           N6F    -0.21392 × 10.sup.-8                                        C-line                                                                               N2C    -0.99652 × 10.sup.-3                                                                   N4C  -0.79680 × 10.sup.-6                           N6C    -0.21619 × 10.sup.-8                                        ______________________________________                                    

                  TABLE 10                                                         ______________________________________                                         (10th Embodiment)                                                              ______________________________________                                         f = 80.9   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    36.031                                                                                  d1      2.82 GRIN lens 1                                        r2    20.673                                                                                  d2     13.15                                                    r3   -30.084                                                                                  d3      3.00 GRIN lens 2                                        r4   -25.638                                                                            Σd = 18.97                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.8050   ν10 = 41.0                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.38397 × 10.sup.-2                                                                   N4d   0.10307 × 10.sup.-4                           N6d    -0.17589 × 10.sup.-7                                        F-line                                                                               N2F    -0.38780 × 10.sup.-2                                                                   N4F   0.10410 × 10.sup.-4                           N6F    -0.17765 × 10.sup.-7                                        C-line                                                                               N2C    -0.38232 × 10.sup.-2                                                                   N4C   0.10262 × 10.sup.-4                           N6C    -0.17514 × 10.sup.-7                                        GRIN lens 2                                                                    N20 = 1.6385   ν20 = 55.6                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.40338 × 10.sup.-3                                                                   N4d  -0.11601 × 10.sup.-7                           N6d    -0.20875 × 10.sup.-8                                        F-line                                                                               N2F    -0.39774 × 10.sup.-3                                                                   N4F  -0.11439 × 10.sup.-7                           N6F    -0.20582 × 10.sup.-8                                        C-line                                                                               N2C    -0.40580 × 10.sup.-3                                                                   N4C  -0.11671 × 10.sup.-7                           N6C    -0.21000 × 10.sup.-8                                        ______________________________________                                    

                  TABLE 11                                                         ______________________________________                                         (11th Embodiment)                                                              ______________________________________                                         f = 80.0   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    43.132                                                                                  d1     3.75  GRIN lens 1                                        r2    44.029                                                                                  d2     8.69                                                     r3   -13.523                                                                                  d3     3.74  GRIN lens 2                                        r4   -19.086                                                                            Σd = 16.18                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.7545   ν10 = 51.6                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.88634 × 10.sup.-3                                                                   N4d  -0.51065 × 10.sup.-6                           N6d     0.22210 × 10.sup.-9                                                                   N8d  -0.26181 × 10.sup.-10                    F-line                                                                               N2F    -0.88944 × 10.sup.-3                                                                   N4F  -0.51244 × 10.sup.-6                           N6F     0.22288 × 10.sup.-9                                                                   N8F  -0.26273 × 10.sup.-10                    C-line                                                                               N2C    -0.88501 × 10.sup.-3                                                                   N4C  -0.50988 × 10.sup.-6                           N6C     0.22177 × 10.sup.-9                                                                   N8C  -0.26142 × 10.sup.-10                    GRIN lens 2                                                                    N20 = 1.5891   ν20 = 61.1                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.20947 × 10.sup.-2                                                                   N4d   0.14782 × 10.sup.-5                           N6d    -0.58140 × 10.sup.-8                                                                   N8d   0.95624 × 10.sup.-10                    F-line                                                                               N2F    -0.21021 × 10.sup.-2                                                                   N4F   0.14834 × 10.sup.-5                           N6F    -0.58344 × 10.sup.-8                                                                   N8F   0.95959 × 10.sup.-10                    C-line                                                                               N2C    -0.20916 × 10.sup.-2                                                                   N4C   0.14760 × 10.sup.-5                           N6C    -0.58053 × 10.sup.-8                                                                   N8C   0.95481 × 10.sup.-10                    ______________________________________                                    

                  TABLE 12                                                         ______________________________________                                         (12th Embodiment)                                                              ______________________________________                                         f = 80.0   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    39.623                                                                                  d1     4.68  GRIN lens 1                                        r2    40.503                                                                                  d2     6.33                                                     r3   -13.324                                                                                  d3     4.23  GRIN lens 2                                        r4   -19.496                                                                            Σd = 15.24                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.5891   ν10 = 61.1                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.78198 × 10.sup.-3                                                                   N4d  -0.11672 × 10.sup.-5                           N6d     0.76312 × 10.sup.-8                                                                   N8d  -0.16338 × 10.sup.-9                     F-line                                                                               N2F    -0.78447 × 10.sup.-3                                                                   N4F  -0.11709 × 10.sup.-5                           N6F     0.76555 × 10.sup.-8                                                                   N8F  -0.16390 × 10.sup.-9                     C-line                                                                               N2C    -0.78091 × 10.sup.-3                                                                   N4C  -0.11656 × 10.sup.-5                           N6C     0.76208 × 10.sup.-8                                                                   N8C  -0.16316 × 10.sup.-9                     GRIN lens 2                                                                    N20 = 1.5891   ν20 = 61.1                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.18907 × 10.sup.-2                                                                   N4d   0.17720 × 10.sup.-5                           N6d    -0.14520 × 10.sup.-7                                                                   N8d   0.22788 × 10.sup.-9                     F-line                                                                               N2F    -0.18985 × 10.sup.-2                                                                   N4F   0.17792 × 10.sup.-5                           N6F    -0.14579 × 10.sup.-7                                                                   N8F   0.22882 × 10.sup.-9                     C-line                                                                               N2C    -0.18874 × 10.sup.-2                                                                   N4C   0.17688 × 10.sup.-5                           N6C    -0.14494 × 10.sup.-7                                                                   N8C   0.22748 × 10.sup.-9                     ______________________________________                                    

                  TABLE 13                                                         ______________________________________                                         (13th Embodiment)                                                              ______________________________________                                         f = 80.0   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    38.933                                                                                  d1     4.49  GRIN lens 1                                        r2    44.339                                                                                  d2     5.99                                                     r3   -13.242                                                                                  d3     4.41  GRIN lens 2                                        r4   -19.031                                                                            Σd = 14.89                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.7130   ν10 = 53.9                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.63960 × 10.sup.-3                                                                   N4d  -0.16052 × 10.sup.-5                           N6d     0.66213 × 10.sup.-8                                                                   N8d  -0.16914 × 10.sup.-9                     F-line                                                                               N2F    -0.64209 × 10.sup.-3                                                                   N4F  -0.16114 × 10.sup.-5                           N6F     0.66471 × 10.sup.-8                                                                   N8F  -0.16980 × 10.sup.-9                     C-line                                                                               N2C    -0.63854 × 10.sup.-3                                                                   N4C  -0.16025 × 10.sup.-5                           N6C     0.66103 × 10.sup.-8                                                                   N8C  -0.16886 × 10.sup.-9                     GRIN lens 2                                                                    N20 = 1.5814   ν20 = 40.9                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.16705 × 10.sup.-2                                                                   N4d   0.16099 × 10.sup.-5                           N6d    -0.18662 × 10.sup.-7                                                                   N8d   0.27776 × 10.sup.-9                     F-line                                                                               N2F    -0.16783 × 10.sup.-2                                                                   N4F   0.16174 × 10.sup.-5                           N6F    -0.18749 × 10.sup.-7                                                                   N8F   0.27906 × 10.sup.-9                     C-line                                                                               N2C    -0.16672 × 10.sup.-2                                                                   N4C   0.16067 × 10.sup.-5                           N6C    -0.18625 × 10.sup.-7                                                                   N8C   0.27720 × 10.sup.-9                     ______________________________________                                    

                  TABLE 14                                                         ______________________________________                                         (14th Embodiment)                                                              ______________________________________                                         f = 80.0   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    18.384                                                                                  d1     2.52  GRIN lens 1                                        r2    10.443                                                                                  d2     5.64                                                     r3   -29.998                                                                                  d3     2.17  GRIN lens 2                                        r4   -16.968                                                                            Σd = 10.33                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.6910   ν10 = 54.8                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.57933 × 10.sup.-2                                                                   N4d   0.26979 × 10.sup.-4                           N6d    -0.29548 × 10.sup.-7                                                                   N8d   0.78201 × 10.sup.-10                    F-line                                                                               N2F    -0.58557 × 10.sup.-2                                                                   N4F   0.27270 × 10.sup.-4                           N6F    -0.29866 × 10.sup.-7                                                                   N8F   0.79043 × 10.sup.-10                    C-line                                                                               N2C    -0.57666 × 10.sup.-2                                                                   N4C   0.26855 × 10.sup.-4                           N6C    -0.29412 × 10.sup.-7                                                                   N8C   0.77840 × 10.sup.-10                    GRIN lens 2                                                                    N20 = 1.6545   ν20 = 33.9                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d     0.20775 × 10.sup.-2                                                                   N4d   0.14042 × 10.sup.-4                           N6d    -0.11763 × 10.sup.-7                                                                   N8d  -0.47234 × 10.sup.-9                     F-line                                                                               N2F     0.21745 × 10.sup.-2                                                                   N4F   0.14697 × 10.sup.-4                           N6F    -0.12312 × 10.sup.-7                                                                   N8F  -0.49438 × 10.sup.-9                     C-line                                                                               N2C     0.20360 × 10.sup.-2                                                                   N4C   0.13761 × 10.sup.-4                           N6C    -0.11528 × 10.sup.-7                                                                   N8C  -0.46289 × 10.sup.-9                     ______________________________________                                    

                  TABLE 15                                                         ______________________________________                                         (15th Embodiment)                                                              ______________________________________                                         f = 80.0   FNo. = 6.5                                                          Radius of  Axial      Refractive Abbe                                          Curvature  Distance   Index (Nd) Number (νd)                                ______________________________________                                         r1    38.813                                                                                  d1     4.39  GRIN lens 1                                        r2    15.300                                                                                  d2     4.16                                                     r3   -29.998                                                                                  d3     3.77  GRIN lens 2                                        r4   -16.967                                                                            Σd = 12.32                                                      ______________________________________                                         GRIN lens 1                                                                    N10 = 1.6968   ν10 = 56.5                                                   (Gradient index coefficients)                                                  d-line                                                                               N2d    -0.34784 × 10.sup.-2                                                                   N4d   0.12825 × 10.sup.-4                           N6d    -0.29111 × 10.sup.-7                                                                   N8d   0.25278 × 10.sup.-8                     F-line                                                                               N2F    -0.35132 × 10.sup.-2                                                                   N4F   0.12953 × 10.sup.-4                           N6F    -0.29402 × 10.sup.-7                                                                   N8F   0.25531 × 10.sup.-9                     C-line                                                                               N2C    -0.34635 × 10.sup.-2                                                                   N4C   0.12770 × 10.sup.-4                           N6C    -0.28986 × 10.sup.-7                                                                   N8C   0.25169 × 10.sup.-9                     GRIN lens 2                                                                    N2 = 1.5891   ν2 = 61.1                                                     (Gradient index coefficients)                                                  d-line                                                                               N2d     0.11611 × 10.sup.-2                                                                   N4d   0.43793 × 10.sup.-5                           N6d     0.31491 × 10.sup.-7                                                                   N8d  -0.45023 × 10.sup.-9                     F-line                                                                               N2F     0.11901 × 10.sup.-2                                                                   N4F   0.44888 × 10.sup.-5                           N6F     0.32278 × 10.sup.-7                                                                   N8F  -0.46149 × 10.sup.-9                     C-line                                                                               N2C     0.11486 × 10.sup.-2                                                                   N4C   0.43324 × 10.sup.-5                           N6C     0.31153 × 10.sup.-7                                                                   N8C  -0.44541 × 10.sup.-9                     ______________________________________                                    

                  TABLE 16                                                         ______________________________________                                                    ν0     ν.sub.Id                                                                           ΣP                                           ______________________________________                                         1st embodiment                                                                              54.8        150    0.014                                          2nd embodiment                                                                              61.1        150    0.062                                          3rd embodiment                                                                              44.9        280    0.053                                          4th embodiment                                                                              46.5        400    0.048                                          ______________________________________                                    

                  TABLE 17                                                         ______________________________________                                         Condition       5th embodiment                                                                             6th embodiment                                     ______________________________________                                         (4)    ν1'       190       200                                                     ν10       51.57     53.93                                                   ν2'       115       100                                                     ν20       54.00     52.85                                            (5)    f1           1.094     1.151                                                   f2                                                                      (6)    ΣP     0.0242    0.0450                                           (7)    ψ1S      -0.393    -0.414                                                  ψ1M                                                                        ψ2S      -0.488    -0.513                                                  ψ2M                                                                 ______________________________________                                    

                  TABLE 18                                                         ______________________________________                                         Condition       7th embodiment                                                                             8th embodiment                                     ______________________________________                                          (8)   ν1'       25        19                                                      ν10       51.6      51.6                                              (9)   f2           2.565     2.281                                                   f                                                                       (10)   ψ2S      -0.828    -0.776                                                  ψ1M                                                                 (11)   ΣP     -0.0013   0.0118                                           (12)   f1           0.523     0.635                                                   f2                                                                      ______________________________________                                    

                  TABLE 19                                                         ______________________________________                                         Condition       9th embodiment                                                                             10th embodiment                                    ______________________________________                                          (4)   ν1'       80        70                                                      ν10       49.77     40.97                                            (13)   ν2'       -95       -50                                              (14)   ν1'       -0.84     -1.40                                                   ν2'                                                                  (15)   f1           1.28      1.07                                                    f2                                                                      (17)   ΣP     0.07      0.08                                             (18)   ψ2S      0.58      1.92                                                    ψ2M                                                                 ______________________________________                                    

                  TABLE 20                                                         ______________________________________                                                 11th         12th       13th                                           Condition                                                                              embodiment   embodiment embodiment                                     ______________________________________                                         (18) ν20 200          170      150                                               ν2'                                                                    (19) f2     1.99         2.19     1.44                                              f                                                                         (20) ψ1S                                                                               0.15         0.13     0.52                                              ψ1M                                                                   (21) ψ2S                                                                               -0.61        -0.65    -0.65                                             ψ2M                                                                   (22) ΣP                                                                              0.0046       0.0083   0.0100                                       (23) f1     0.82         0.69     0.61                                              f2                                                                        ______________________________________                                    

                  TABLE 21                                                         ______________________________________                                         Condition      14th embodiment                                                                             15th embodiment                                    ______________________________________                                         (24)   ν1'      65         70                                                      ν10      54.75      56.47                                            (25)   ν2'      15         28                                                      ν20      33.86      61.11                                            (26)   ψ2S     -1.98      -1.27                                                   ψ2M                                                                 (27)   ψ1S     -0.85      -0.88                                                   ψ1M                                                                 (28)   ΣP    0.13       0.18                                             (29)   f1          1.78       1.42                                                    f2                                                                      ______________________________________                                     

What is claimed is:
 1. A radial gradient index lens system having a refractive index distribution where a refractive index decreases from an optical axis along a height, said gradient index lens system being of a completely symmetrical type, about an aperture stop, comprisingtwo meniscus lens elements having a refractive index distribution where a refractive index decreases from an optical axis along a height, whose convex surfaces have a larger radius of curvature than concave surfaces, said meniscus lens elements being arranged symmetrically with respect to the aperture stop so that the convex surfaces thereof face an object side and an image side, respectively, wherein both of said two meniscus lens elements fulfill the following condition:

    ν<ν 1d

whereν0 represents an axial Abbe number of the lens element; and where ν1d is equal to ##EQU13## where N1d represents a secondary gradient index coefficient of d-line, N1F represents a secondary gradient index coefficient of F-line and N1C represents a secondary gradient index coefficient of C-line.
 2. A radial gradient index lens system as claimed in claim 1, wherein said gradient index lens system is for use in a copying apparatus and has a projection capability characteristic of a letter-size document.
 3. A radial gradient index lens system as claimed in claim 1, wherein the following conditions are fulfilled:

    0.20>ΣP>-0.05

where: ΣP represents a Petzval sum of the lens system.
 4. A radial gradient index lens system as claimed in claim 3, wherein said gradient index lens system is for use in a copying apparatus.
 5. A radial gradient index lens system comprising two lens elements having a refractive power distribution from an optical axis along a height, said two lens elements consisting of, from the object side, a first lens element where a power of lens configuration is positive and a power of lens material is positive and a total power is positive, and a second lens element where a power of lens configuration is negative and a power of lens material is positive a total power is positive, wherein the following conditions are fulfilled: ##EQU14## where ν20represents an axial Abbe number of a second lens element;ν20' is calculated by ##EQU15## where N2d represents a secondary gradient index coefficient of d-line of the second lens element, N2F represents a secondary gradient index coefficient of F-line of the second lens element and N2C represents a secondary gradient index coefficient of C-line of the second lens element; F represents a focal length of an entire lens system (total of the lens configuration and lens material); f1 represents a focal length of a first lens element (total of the lens configuration and lens material); f2 represents a focal length of the second lens element (total of the lens configuration and lens material); ψ2S represents the power of lens configuration of the second lens element; ψ2M represents the power of lens material of the second lens element; ΣP represents a Petzval sum of the entire lens system; ψ1S represents the power of lens configuration of the first lens element; and ψ1M represents the power of lens material of the first lens element.
 6. A radial gradient index lens system as claimed in claim 5, wherein said gradient index lens system is for use as a reading optical system.
 7. A radial gradient index lens system comprising two lens elements having a refractive power distribution from an optical axis along a height, said two lens elements consisting of, from the object side, a first lens element where a power of lens configuration is negative and a power of lens material is positive and a total power is positive, and a second lens element where a power of lens configuration is positive and a power of lens material is negative and a total power is positive, wherein the following conditions are fulfilled: ##EQU16## where: ν10represents an axial Abbe number of the first lens element;ν20 represents an axial Abbe number of the second lens element; ν1' is equal to ##EQU17## where N_(2d) ¹ represents a secondary gradient index coefficient of d-line of the first lens element, N_(2F) ¹ represents a secondary gradient index coefficient of F-line of the first lens element and N_(2C) ¹ represents a secondary gradient index coefficient of C-line of the first lens element; ν' is equal to ##EQU18## where N_(2d) ² represents a secondary gradient index coefficient of d-line of the second lens element, N_(2F) ² represents a secondary gradient index coefficient of F-line of the second lens element and N_(2C) ² represents a secondary gradient index coefficient of C-line of the second lens element; f represents a focal length of an entire lens element (total of the lens configuration and lens material); f1 represents a focal length of the first lens element (total of the lens configuration and lens material); f2 represents a focal length of the second lens element (total of the lens configuration and lens material); ψ2S represents the power of lens configuration of the second lens element; ψ2M represents the power of lens material of the second lens element; ΣP represents a Petzval sum of the entire lens system; ψ1S represents the power of lens configuration of the first lens element; and ψ1M represents the power of lens medium of the first lens element.
 8. A radial gradient index lens system as claimed in claim 7, wherein said gradient index lens system is for use as a reading optical system.
 9. A radial gradient index lens system consisting of two lens elements having a refractive index distribution from an optical axis along a height, wherein in both of the two lens elements, a power of lens configuration is negative and a power of lens material is positive, wherein both of the two lens elements fulfill the following condition:

    νi0<νi'

where νi0 represents an axial Abbe number of an i-th lens element; and νi' is calculated by ##EQU19## where N_(2F) ^(i) represents a secondary gradient coefficient of F-line of the i-th lens element, N_(2C) ^(i) represents a secondary gradient index coefficient of C-line of the i-th lens element, N_(2d) ^(i) represents a secondary gradient index coefficient of d-line of the i-th lens element.
 10. A radial gradient index lens system as claimed in claim 9, wherein said gradient index lens system is for use as a reading optical system.
 11. A radial gradient index lens system as claimed in claim 9, wherein the following conditions are further fulfilled: ##EQU20## where f1 represents a focal length of the first lens element (total of the lens configuration and lens material), f2 represents a focal length of the second lens element (total of the lens configuration and lens material), ψiS represents to power of lens configuration of the ith lens element, ψiM represents the power of lens material of the ith lens element, and ΣP represents a Petzval sum of the entire lens system.
 12. A radial gradient index lens system as claimed in claim 11, wherein said gradient index lens system is for use as a reading optical system.
 13. A radial gradient index lens system comprising two lens elements having a refractive index distribution from an optical axis along a height, said two lens elements consisting of, from the object side, of a first lens element where a power of lens configuration is positive and a power of lens material is negative and a total power is positive, and a second lens element where a power of lens configuration is negative and a power of lens material is positive and a total power is positive, wherein both of the two lens elements fulfill the following condition:

    0<ν1'<ν10

    ν20<ν2'

where ν10 represents an axial Abbe number of a first lens element and ν20 represents an axial Abbe number of a second lens element; and ν1' and ν2' are calculated by the general equation ##EQU21## where N_(2F) ^(i) represents a secondary gradient index coefficient of F-line of the i-th lens element, N_(2C) ^(i) represents a secondary gradient index coefficient of C-line of the i-th lens element, N_(2d) ^(i) represents a secondary gradient index coefficient of d-line of the i-th lens element where the i-th element can be either the first or second lens element.
 14. A radial gradient index lens system as claimed in claim 13, wherein said gradient index lens is for use as a reading optical system.
 15. A radial gradient index lens system comprising two lens elements having a refractive index distribution where a refractive index decreases from an optical axis along a height, said two lens elements consisting of, from the object side, a first lens element where a power of lens configuration is negative and a power of lens material is positive, and a second lens element where both of a power of lens configuration and a power of lens material is positive, wherein both of the two lens elements fulfill the following condition:

    ν10<ν1'

    ν2'<0

ν10represents an axial Abbe number of the first lens element; ν1' and ν2' are calculated by ##EQU22## where N_(2F) ^(i) represents a secondary gradient index coefficient of F-line of the i-th lens element, N_(2C) ^(i) represents a secondary gradient index coefficient of C-line of the i-th lens element, N_(2d) ^(i) represents a secondary gradient index coefficient of d-line of the i-th lens element.
 16. A radial gradient index lens system as claimed in claim 15, wherein said gradient index lens system is for use as a reading optical system.
 17. A radial gradient index lens system as claimed in claim 15, wherein the following conditions are fulfilled: ##EQU23## where νi0 represents an axial Abbe number of an ith lens element, νi' is calculated by ##EQU24## where N_(2d) ^(i) represents a secondary gradient index coefficient of d-line of the ith lens element, N_(2F) ^(i) represents a secondary gradient index coefficient of F-line of the ith lens element and N_(2C) ^(i) represents a secondary gradient index coefficient of C-line of the ith lens element.
 18. A radial gradient index lens system as claimed in claim 17, wherein said gradient index lens system is for use as a reading optical system.
 19. A radial gradient index lens system as claimed in claim 17, wherein the following conditions are further fulfilled: ##EQU25## where f1 represents a focal length of the first lens element, f2 represents a focal length of the second lens element, ΣP represents a Petzval sum of the entire lens system, ψ2S represents the power of lens configuration of the second lens element, and ψ2M represents the power of lens material of the second lens element.
 20. A radial gradrient index lens system as claimed in claim 19, wherein said gradient index lens system is for use as a reading optical system.
 21. A radial gradient index lens system comprising two lens elements having a refractive index distribution from an optical axis along a height, wherein in a first lens element, a power of the lens configuration is positive and a power of the lens material is negative, wherein in a second lens element, a power of the lens configuration is negative and a power of the lens material is positive, and wherein the following condition is fulfilled:

    0<ν1'<ν10

where ν1' is calculated by ##EQU26## where N2d represents a secondary gradient index coefficient of d-line of a first lens element, N2F represents a secondary gradient index coefficient of F-line of the first lens element, N2C represents a secondary gradient index coefficient of C-line of the first lens element, and ν10 represents an axial Abbe number of the first lens element.
 22. A radial gradient index lens system as claimed in claim 21, wherein said gradient index lens system is for use as a reading optical system.
 23. A radial gradient index lens system as claimed in claim 21, wherein the following conditions are further fulfilled: ##EQU27## where f represents a focal length of an entire lens system, f1 represents a focal length of the first lens element (total of the lens configuration and lens material), f2 represents a focal length of a second lens element (total of the lens configuration and lens material), ψ2S represents the power of the lens configuration of the second lens element, ψ2M represents the power of lens material of the second lens element, and ΣP represents a Petzval sum.
 24. A radial gradient index lens system as claimed in claim 23, wherein said gradient index lens system is for use as a reading optical system. 